1. Lecture 4: Sampling [2] XILIANG LUO 2014/10

2. Periodic Sampling  A continuous time signal is sampled periodically to obtain a discrete- time signal as: Ideal C/D converter

3. Ideal Sampling  Impulse train modulator

4. Fourier Transform of Ideal Sampling Fourier Transform of periodic impulse train is an impulse train:

5. What about DTFT This is the general relationship between the periodically sampled sequence and the underlying continuous time signal

6. Nyquist-Shannon Sampling

7. Process Cont. Signal  A main application of discrete-time systems is to process continuous- time signal in discrete-time domain

8. Band-limited Signal

9. Observations  For band-limited signal, we are processing continuous time signal using discrete-time signal processing  For band-limited signal, the overall system behaves like a linear time- invariant continuous-time system with the following frequency domain relationship:

10. Process Discrete-Time Signal

12. Example: Non-Integer Delay

13. HW Due on 10/10 4.31 4.34 4.53 4.60 4.61 4.21 4.54 need multi-rate signal processing knowledge

14. Next 1. Change sampling rate 2. Multi-rate signal processing 3. Quantization 4. Noise shaping

15. Change Sampling Rate Conceptually, we can do this by reconstruct the continuous time signal first, then resample the reconstructed continuous signal

16. Sampling Rate Reduction Down-sampling

17. Downsampling

19. Anti-Aliasing Filter

20. Aliasing Example

21. Upsampling

23. Frequency Domain

24. Upsampling

25. Filtering   Compressor

26. Filtering   Expander

27. Polyphase Decomposition Goal: efficient implementation structure k=0,1,…,M-1

28. Polyphase Decomposition

29. Polyphase in Freq Domain Polyphase component filters

30. Polyphase Filters y[n]=x[n]*h[n]

31. Polyphase + Decimation Filter

34. Polyphase + Interp Filter

37. Ideal

38. Practical

39. Avoid Aliasing

40. Simple Anti-Aliasing Filter

41. Oversampling  C/D

43.  Advantages  nominal analog filter  exact linear phase

44. A/D Conversion

45. Zero-order Hold System

46. Quantization

47. a Typical Quantizer

48. Quantization Error

49. Assumptions:

50. Quantization Error

51. D/A Conversion Ideal reconstruction:

52. D/A Conversion

53. Effect of Quantization:

54. D/A Conversion

55. compensated filter

56. D/A Conversion

59. Practical D/A Conversion

60. Practical Digital System